263 



Substituting this value in equation (6A): 



. _ (Bibi sin Xi-\-B2b2 sin ^2+ • • -T^ 

 ^ Aa^ -\-B lb i^ cos Xi-\- Bibo^ cos X2-\- ' • • 



_Bi~bi^sm'^Xi-{-B2%^sm^ X2-\~ • -h'2>BiB2bib2sinxisinx2-{- • •• 



~ Aa:'+B,b{^ cos Xi+B2b2' cos X2+ • • • (12A) 



6. Mean value of Ay. — The symbols Xi, X2, etc., in equation (12 A); 

 represent angles in the form (equation 5A): 



x=2nTbla—bala-{' 13 

 where n is an integer. 



As successive integral values are assigned to n, x increases by: 



27r6/a=27r(6-a)/a+27r. 



At each increase in n, the value of x increases, therefore by 

 2ir{b — a)la. As the speed, 6, of any sem.idiurnal com.ponent does not 

 differ greatly from, a, the speed of the dominant com.ponent, the 

 fraction 27r {b—a)/a is com.paratively small for such components. 

 The successive values of x steadily increase (or decrease) with each 

 increase in n by an angle which describes a small fraction of the cir- 

 cumference. The speeds of the diurnal components (except Mi) 

 differ by a relatively small am.ount from, one-half of that of the dom.i- 

 nant com.ponent M2. For these components the value of x steadily 

 increases by a little m.ore or less than 180° with each increase in n. 

 In either case the values of x fall uniformly, in the long run, over the 

 entire range of angles from. to 27r, and the mean values of the trigono- 

 m.etric functions of x in equation (12A) become their true m.ean 

 values as x varies from to 2t. The mean value of sin ^x between 

 these limits is one-half, while that of cos .T,and of the products of the sines 

 of the differently varying angles Xi, X2, etc., is zero. Aside then from 

 the effects of the Mi component and the lunar overtides, the mean 

 value of Ay, becomes: 



Ayo=}i{B,\'+B.32'+ ■ • •)fAa' (13A) 



7. Mean high water in terms of the harmonic components. — Since the 

 height of m.ean high water above m.ean sea level is the amplitude of 

 the dom.inant component increased by one-half of the mean value of 

 Ay, it is given by the expression: 



MRW=Ai-%iB,%'+B.2'h^'+ ■ ■ ■)IAa' 



=A[l + }i(B3i'/A'a'+B./b2yA'a'+ ■ • •) (14A) 



